Properties

Label 2-48-16.11-c8-0-15
Degree $2$
Conductor $48$
Sign $0.513 + 0.857i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.8 + 11.7i)2-s + (−33.0 − 33.0i)3-s + (−18.7 − 255. i)4-s + (260. + 260. i)5-s + (747. − 27.4i)6-s − 368.·7-s + (3.19e3 + 2.56e3i)8-s + 2.18e3i·9-s + (−5.88e3 + 215. i)10-s + (−1.70e4 + 1.70e4i)11-s + (−7.82e3 + 9.06e3i)12-s + (3.22e4 − 3.22e4i)13-s + (4.01e3 − 4.32e3i)14-s − 1.72e4i·15-s + (−6.48e4 + 9.58e3i)16-s − 1.20e5·17-s + ⋯
L(s)  = 1  + (−0.680 + 0.732i)2-s + (−0.408 − 0.408i)3-s + (−0.0733 − 0.997i)4-s + (0.416 + 0.416i)5-s + (0.576 − 0.0211i)6-s − 0.153·7-s + (0.780 + 0.625i)8-s + 0.333i·9-s + (−0.588 + 0.0215i)10-s + (−1.16 + 1.16i)11-s + (−0.377 + 0.437i)12-s + (1.12 − 1.12i)13-s + (0.104 − 0.112i)14-s − 0.339i·15-s + (−0.989 + 0.146i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.513 + 0.857i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.648265 - 0.367428i\)
\(L(\frac12)\) \(\approx\) \(0.648265 - 0.367428i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.8 - 11.7i)T \)
3 \( 1 + (33.0 + 33.0i)T \)
good5 \( 1 + (-260. - 260. i)T + 3.90e5iT^{2} \)
7 \( 1 + 368.T + 5.76e6T^{2} \)
11 \( 1 + (1.70e4 - 1.70e4i)T - 2.14e8iT^{2} \)
13 \( 1 + (-3.22e4 + 3.22e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 1.20e5T + 6.97e9T^{2} \)
19 \( 1 + (-5.00e3 - 5.00e3i)T + 1.69e10iT^{2} \)
23 \( 1 - 4.28e5T + 7.83e10T^{2} \)
29 \( 1 + (-6.69e5 + 6.69e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 3.56e4iT - 8.52e11T^{2} \)
37 \( 1 + (1.95e6 + 1.95e6i)T + 3.51e12iT^{2} \)
41 \( 1 + 2.75e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.02e6 + 2.02e6i)T - 1.16e13iT^{2} \)
47 \( 1 + 5.13e6iT - 2.38e13T^{2} \)
53 \( 1 + (2.06e6 + 2.06e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (9.48e6 - 9.48e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-8.73e6 + 8.73e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.48e7 + 1.48e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 2.71e7T + 6.45e14T^{2} \)
73 \( 1 - 7.57e6iT - 8.06e14T^{2} \)
79 \( 1 + 5.69e6iT - 1.51e15T^{2} \)
83 \( 1 + (-7.10e6 - 7.10e6i)T + 2.25e15iT^{2} \)
89 \( 1 + 9.01e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.12e8T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.73733306775953381165554480393, −12.82791717826190222576202887140, −10.89843153629122166473142939542, −10.24051929601060909248380351980, −8.668956982628868341320705578890, −7.36734273467067432005675232842, −6.31221124141316058259107319164, −5.04205887541623715556250522115, −2.22478300419098740678363555892, −0.39918719254694287292169113853, 1.21587529028219119817414043210, 3.09159879540386489420849693509, 4.79072103203124592229625246037, 6.56645767604862400791127874873, 8.509305513857228272370705112577, 9.268656062781086344900793476246, 10.79498581611966747536649675454, 11.28678562291832862631877249707, 12.93519787830919185432272183593, 13.63379432623191148236851416392

Graph of the $Z$-function along the critical line